The **Chomsky hierarchy** is a containment hierarchy of classes of formal grammars that generate formal languages. This hierarchy was described by Noam Chomsky in 1956.

Table of contents |

2 The hierarchy 3 References |

### Formal grammars

and start symbol S, defines the language of all words of the form (i.e.*n*copies of

*a*followed by

*n*copies of

*b*).

See formal grammar for a more elaborate explanation.

### The hierarchy

The Chomsky hierarchy consists of the following levels:

- Type-0 grammars (unrestricted grammars) include all formal grammars. They generate exactly all languages that can be recognized by a Turing machine. The language that is recognized by a Turing machine is defined as all the strings on which it halts. These languages are also known as the recursively enumerable languages. Note that this is different from the recursive languages which can be
*decided*by an always halting Turing machine. - Type-1 grammars (context-sensitive grammars) generate the context-sensitive languages. These grammars have rules of the form α
*A*β -> αγβ with*A*a nonterminal and α, β and γ strings of terminals and nonterminals. The strings α and β may be empty, but γ must be nonempty. The rule S -> ε is allowed if S does not appear on the right side of any rule. The languages described by these grammars are exactly all languages that can be recognized by a non-deterministic Turing machine whose tape is bounded by a constant times the length of the input. - Type-2 grammars (context-free grammars) generate the context-free languages. These are defined by rules of the form
*A*-> γ with*A*a nonterminal and γ a string of terminals and nonterminals. These languages are exactly all languages that can be recognized by a non-deterministic pushdown automaton. Context free languages are the theoretical basis for the syntax of most programming languages. - Type-3 grammars (regular grammars) generate the regular languages. Such a grammar restricts its rules to a single nonterminal on the left-hand side and a right-hand side consisting of a single terminal, possibly followed by a single nonterminal. The rule S -> ε is also here allowed if S does not appear on the right side of any rule. These languages are exactly all languages that can be decided by a finite state automaton. Additionally, this family of formal languages can be obtained by regular expressions. Regular languages are commonly used to define search patterns and the lexical structure of programming languages.

The following table summarizes each of Chomsky's four types of grammars, the class of languages it generates, the type of automaton that recognizes it, and the form its rules must have.

Grammar | Languages | Automaton | Production rules |
---|---|---|---|

Type-0 | Recursively enumerable | Turing machine | No restrictions |

Type-1 | Context-sensitive | Linear-bounded non-deterministic Turing machine | αAβ -> αγβ |

Type-2 | Context-free | Non-deterministic pushdown automaton | A -> γ |

Type-3 | Regular | Finite state automaton | A -> aBA -> a |